Optimal Large-time Behavior of the Vlasov-maxwell-boltzmann System in the Whole Space

In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space R. The existence of global in time nearby Maxwellian solutions is known from [37] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work [12] on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t− 3 2 + 3 2r ) in L ξ (Lx)-norm for any 2 ≤ r ≤ ∞ if initial perturbation is smooth enough and decays in space-velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided.